Companion-Rule-List (CRL): Modal Companion Rules

• Companion-Rule-List (CRL) is a canonical set of modal stable-canonical rules derived from superintuitionistic rule-systems using finite filtration and Gödel translation.
• The CRL framework extends the Blok–Esakia correspondence by mapping si–rules to modal counterparts via finite bounded distributive lattice embeddings.
• Its algorithmic construction and uniform extension to bi-intuitionistic, tense, and polymodal logics offer a semantically transparent method for analyzing rule-system properties.

A Companion-Rule-List (CRL) is a canonical set of modal rules systematically associated to a given superintuitionistic rule-system (si–rule-system) through the machinery of stable (and pre-stable) canonical rules. This association extends the Blok–Esakia Galois connection from logics to the level of rule-systems, offering a uniform, semantically transparent mechanism for capturing modal companions, refutation patterns, and correspondence phenomena across a wide spectrum of logical signatures—including intuitionistic, modal, bi-intuitionistic, and tense logics (Bezhanishvili et al., 2022).

1. Formal Definition and Construction

Given a superintuitionistic rule-system $R$ over some signature $\text{si}$, the Companion-Rule-List $\operatorname{CRL}(R)$ is defined using the following construction:

1. Gödel Translation: For each inference rule $\Gamma/\Delta$ in $R$, apply the Gödel translation $T(\Gamma/\Delta)$ to map formulas into the modal language (standard translation for S4).
2. Modal System Extension: Define $$\tau(R) := S4_R \oplus \{ T(\Gamma/\Delta) : \Gamma/\Delta \in R \}$$, where $S4_R$ is the base modal rule-system.
3. Stable-Canonical Rules: $\operatorname{CRL}(R)$ is then the unique set of modal stable-canonical rules whose universal-class closure equals $\tau(R)$. Each such rule can be written as $\text{si}$0 for a finite S4-algebra $\text{si}$1 and parameter set $\text{si}$2:

$\text{si}$3

Lemma 3.12 ensures that every si-rule $\text{si}$4 is equivalent (over $\text{si}$5) to finitely many stable-canonical rules (Bezhanishvili et al., 2022).

2. Filtration, Embeddings, and Stable-Canonical Blueprints

The methodology behind CRL construction leverages finite filtration and stable-canonical rule encoding:

• Every non-valid si-rule $\text{si}$6 in a Heyting algebra $\text{si}$7 is “filtered” to a finite bounded distributive lattice $\text{si}$8 with an expansion $\text{si}$9 yielding a bounded-lattice embedding $\operatorname{CRL}(R)$0 that satisfies the Bounded-Domain-Condition (BDC) for the relevant parameters.
• The finite refutation pattern $\operatorname{CRL}(R)$1 is captured by a stable-canonical rule $\operatorname{CRL}(R)$2:

$\operatorname{CRL}(R)$3

• Propositions 3.10–3.11 establish that $\operatorname{CRL}(R)$4 refutes $\operatorname{CRL}(R)$5 iff there exists a bounded-lattice embedding of $\operatorname{CRL}(R)$6 into $\operatorname{CRL}(R)$7 satisfying the BDC for $\operatorname{CRL}(R)$8; in the dual Esakia-space semantics, this corresponds to a continuous surjection fulfilling the dual BDC (Bezhanishvili et al., 2022).

3. Rule-Level Blok–Esakia Correspondence

A central result is the extension of the Blok–Esakia theorem from logics to rule-systems:

• The maps

$\operatorname{CRL}(R)$9

defined by

$\Gamma/\Delta$0

form a Galois connection, which for $\Gamma/\Delta$1-extensions is a lattice isomorphism. Specifically,

$\Gamma/\Delta$2

with the property that, for any si–rule–system $\Gamma/\Delta$3,

$\Gamma/\Delta$4

Modal companions of $\Gamma/\Delta$5 are exactly those $\Gamma/\Delta$6 satisfying $\Gamma/\Delta$7 (Bezhanishvili et al., 2022).

4. Illustrative Examples

Three key cases demonstrate the breadth of the Companion-Rule-List construction and its correspondence properties:

Case	Rule-System	CRL Description
(a) IPC → S4	$\Gamma/\Delta$8 (minimal si)	$\Gamma/\Delta$9 yields S4-canonical rules; reflexivity and transitivity axioms via stable-canonical rules $R$0, $R$1
(b) KM → GL	$R$2 (minimal modal-si)	$R$3 recovers GL-canonical rules; uses pre-stable canonical rules derived from finite frontons, with each $R$4–rule rewritten into pre-stable rules $R$5
(c) Bi-intuitionistic / tense logics	Bi-IPC, Tense signatures	Combine two stable-canonical rules for (→, co→), or for tense (two box modalities); yields an isomorphism $R$6

Each example follows the uniform recipe: filtration → finite countermodel → canonical rule encoding → aggregation into the CRL.

5. General Recipe and Algorithmic Construction

A constructive “pseudocode” approach realizes $R$7 from a finite si–rule–system presentation:

$R$9 Mutatis mutandis for modal/fronton cases (requiring Boolean local finiteness).

6. Extensions to Richer Logical Signatures

The Companion-Rule-List methodology extends uniformly to a variety of logical languages and structural enrichments:

• Bi-implication: Add a second filtration parameter for the co-implication connective, extending the stable-canonical rule grammar.
• Tense / Hybrid logics: Introduce two BDC-governed modal parameters (future-box, past-box), generalizing the filtration and encoding process.
• Many-sorted / Polymodal logics: As long as the signature supports locally finite filtration, any non-valid rule admits finite canonicalization into CRL form.

The overarching construction in all cases comprises four steps: select a filtration notion; certify finite countermodels; encode via stable/pres-stable canonical rules; and aggregate the resulting rules to form $R$8. This delivers a unified account of rule-level Blok–Esakia, the Dummett–Lemmon conjecture, and the Kuznetsov–Muravitsky isomorphism (Bezhanishvili et al., 2022).

7. Significance and Theoretical Implications

The Companion-Rule-List framework provides a powerful and semantically transparent mechanism for:

• Lifting classical modal–intuitionistic correspondences to the rule-syntactic level.
• Enabling effective axiomatizations in terms of stable and pre-stable canonical rules for any rule-system admitting local finiteness via filtration.
• Clarifying the algebraic and topological dualities inherent in the structure of modal companions, not only for basic intuitionistic and modal logics, but for bi-intuitionistic, tense, and polymodal extensions.
• Systematizing the construction of modal companions and establishing bijective Galois connections in generalized settings, connecting algebraic, syntactic, and semantic perspectives within non-classical logic research (Bezhanishvili et al., 2022).

A plausible implication is that this approach will facilitate further advances in the automated generation of canonical rules for newly proposed logical systems where variations of filtration and stable-canonical construction remain applicable.